Local Theorems for Arithmetic Multidimensional Compound Renewal Processes under Cramér’s Condition

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Local Theorems for Arithmetic Multidimensional Compound Renewal Processes under Cramér’s Condition. / Mogul’skiĭ, A. A.; Prokopenko, E. I.

In: Siberian Advances in Mathematics, Vol. 30, No. 4, 11.2020, p. 284-302.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{6c90e249bb614587b253ae80da43e340,

title = "Local Theorems for Arithmetic Multidimensional Compound Renewal Processes under Cram{\'e}r{\textquoteright}s Condition",

abstract = "We continue the study of compound renewal processes (c.r.p.) under Cram{\'e}r{\textquoteright}smoment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensionalc.r.p. Z(n) and Y(n), for which the random vector ξ = (τ, ζ) controlling these processes (τ > 0 defines the distance between jumps, ζ defines the value of jumps of the c.r.p.)has an arithmetic distribution and satisfies Cram{\'e}r{\textquoteright}s moment condition. For theseprocesses, we find the exact asymptotics in the local limit theorems for the probabilities P (Z(n) = x), P (Y(n) = x) in theCram{\'e}r zone of deviations for x ∈ Zd (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p.,where the vector ξ = (τ, ζ) has a nonlattice distribution).",

keywords = "arithmetic distribution, compound renewal process, Cram{\'e}r{\textquoteright}s condition, deviations function, large deviations, local limit theorem, moderate large deviations, renewal function",

author = "Mogul{\textquoteright}skiĭ, {A. A.} and Prokopenko, {E. I.}",

year = "2020",

month = nov,

doi = "10.1134/S1055134420040033",

language = "English",

volume = "30",

pages = "284--302",

journal = "Siberian Advances in Mathematics",

issn = "1055-1344",

publisher = "PLEIADES PUBLISHING INC",

number = "4",

}

RIS

TY - JOUR

T1 - Local Theorems for Arithmetic Multidimensional Compound Renewal Processes under Cramér’s Condition

AU - Mogul’skiĭ, A. A.

AU - Prokopenko, E. I.

PY - 2020/11

Y1 - 2020/11

N2 - We continue the study of compound renewal processes (c.r.p.) under Cramér’smoment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensionalc.r.p. Z(n) and Y(n), for which the random vector ξ = (τ, ζ) controlling these processes (τ > 0 defines the distance between jumps, ζ defines the value of jumps of the c.r.p.)has an arithmetic distribution and satisfies Cramér’s moment condition. For theseprocesses, we find the exact asymptotics in the local limit theorems for the probabilities P (Z(n) = x), P (Y(n) = x) in theCramér zone of deviations for x ∈ Zd (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p.,where the vector ξ = (τ, ζ) has a nonlattice distribution).

AB - We continue the study of compound renewal processes (c.r.p.) under Cramér’smoment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensionalc.r.p. Z(n) and Y(n), for which the random vector ξ = (τ, ζ) controlling these processes (τ > 0 defines the distance between jumps, ζ defines the value of jumps of the c.r.p.)has an arithmetic distribution and satisfies Cramér’s moment condition. For theseprocesses, we find the exact asymptotics in the local limit theorems for the probabilities P (Z(n) = x), P (Y(n) = x) in theCramér zone of deviations for x ∈ Zd (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p.,where the vector ξ = (τ, ζ) has a nonlattice distribution).

KW - arithmetic distribution

KW - compound renewal process

KW - Cramér’s condition

KW - deviations function

KW - large deviations

KW - local limit theorem

KW - moderate large deviations

KW - renewal function

UR - http://www.scopus.com/inward/record.url?scp=85095950703&partnerID=8YFLogxK

U2 - 10.1134/S1055134420040033

DO - 10.1134/S1055134420040033

M3 - Article

AN - SCOPUS:85095950703

VL - 30

SP - 284

EP - 302

JO - Siberian Advances in Mathematics

JF - Siberian Advances in Mathematics

SN - 1055-1344

IS - 4

ER -

ID: 25865548